A210238 Triangle of multiplicities D(n) of multinomial coefficients corresponding to sequence A210237.

1, 2, 1, 3, 6, 1, 4, 12, 6, 12, 1, 5, 20, 20, 30, 30, 20, 1, 6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1, 7, 42, 42, 42, 105, 210, 105, 245, 420, 140, 105, 210, 42, 1, 8, 56, 56, 224, 28, 336, 336, 280, 168, 168, 840, 420, 1120, 70, 1120, 560, 168, 420, 56, 1

Offset

1,2

Comments

Multiplicity D(n) of multinomial coefficient M(n) is the number of ways the same value of M(n)=n!/(m1!*m2!*..*mk!) is obtained by distributing n identical balls into k distinguishable bins.

Differs from A209936 after a(21).

Differs from A035206 after a(36).

The checksum relationship: sum(M(n)*D(n)) = k^n

The number of terms per row (for each value of n starting with n=1) forms sequence A070289.

Links

Table of n, a(n) for n=1..63.

Sergei Viznyuk, C-program for the sequence.

Example

1
2, 1
3, 6, 1
4, 12, 6, 12, 1
5, 20, 20, 30, 30, 20, 1
6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1
7, 42, 42, 42, 105, 210, 105, 245, 420, 140, 105, 210, 42, 1
Thus for n=3 (third row) the same value of multinomial coefficient follows from the following combinations:
3!/(3!0!0!) 3!/(0!3!0!) 3!/(0!0!3!) (i.e. multiplicity=3)
3!/(2!1!0!) 3!/(2!0!1!) 3!/(0!2!1!) 3!/(0!1!2!) 3!/(1!0!2!) 3!/(1!2!0!)  (i.e. multiplicity=6)
3!/(1!1!1!) (i.e. multiplicity=1)

Mathematica

Table[Last/@Tally[Multinomial@@@Compositions[k, k]], {k, 8}] (* Wouter Meeussen, Mar 09 2013 *)

Crossrefs

Cf. A210237, A209936, A035206, A070289.

Sequence in context: A309220 A225632 A035206 * A209936 A213941 A360858

Adjacent sequences: A210235 A210236 A210237 * A210239 A210240 A210241

Keyword

nonn,tabf

Author

Sergei Viznyuk, Mar 18 2012

Status

approved